Scalar Curvature in Seiberg-Witten Theory
Yong Seung Cho
Department of Mathematics, Ewha Women’s University, Seoul 120-750, Korea e-mail : [email protected]
(2000 Mathematics Subject Classification : 53C40, 53C15.)
Abstract. In this short survey article, we would like to introduce scalar curvature in four- manifold, Seiberg-Witten invariant, and some applications of Seiberg-Witten invariant.
1 Short Review of Riemannian Geometry
Let (M, g) be a Riemannian manifold. There is a unique torsion free Riman- nian connection ∇ on (M, g) which is called the Levi-Civita connection of g. The curvature operator R of (M, g) is the curvature tensor field R of g, a section of End(∧2T M) ;
< R(W ∧X), Y ∧Z >∧2T M=< R(W, X)Z, Y >T M .
The sectional curvature of (M, g) with respect to a 2-planeσ⊂TpM,p∈M, is Kσ=Ku∧v =<R(u∧v),u∧v>
ku∧vk2
where uandv∈TpM span the planeσ.
The Ricci curvature of (M, g) with respect to a nonzero vector v∈T M is γ(v) = tr(R(•,v)v)
kvk2 .
The scalar curvature of (M, g) is the maps:M →Rgiven by
s(p) = Xn
i=1
γ(ei) =X
i,j
< R(ei, ej), ej> =X
i6=j
Kei∧ej = 2X
i<j
Kei∧ej,
where {ei}i=1,···,n is an orthonormal basis forTpM. Now we introduce Weitzenb¨ock’s Formulas :
The divergence of a (0, s)-tensor field φonM, s >0, is the (0, s−1)-tensor field divφonM such that at p∈M,
177
(divφ)p:=P
ivei∇eiφ 1) (divφ)p(v1,· · · , vs−1) =P
i∇eiφ(ei, v1,· · ·, vs−1) 2) (divX)p=tr[(∇X)p] =P
iwi(∇eiX)
where X is a vector field,{ei} is a basis(orthonormal) of TpM and {wi} the dual basis.
3) (divf)p= 0 for a functionf :M →R.
(1.1) Ifφis a differential form on (M, g), then 4φ=−div∇φ+ρφ,
<4φ, φ >=124kφk2+k∇φk2+< ρφ, φ >
Ifφis ar-form, then ρφ(v1,· · ·, vr) =Pn
i=1
Pr
j=1(R(ei, vj)φ)(v1,· · · , vj−1, ei, vj+1,· · ·, vr).
(1.2) The exact sequence
0→Z2→spin(n)→SO(n)→0 yields a long exact cohomology sequence :
→H1(M,Z2)→H1(M, spin(n))β→H1(M, SO(n))→H2(M,Z2)→ · · ·. For 1- ˇCech cocycle g of a spin structure on M : w2 = 0 iffg=β(˜g) for some
˜
g∈H1(M,spin(n)).The distinct spin structures onM are in one-to-one correspon- dence withH1(M,Z2). For example, forn≥2 the sphereSn admits a unique spin structure, the torusS1×S1 admits 4 distinct spin structures.
The Dirac operator of a given spin structure onM is the first order differential operator D : Γ(W) → Γ(W) such that Dφ = m· ∇φ where m is the Clifford multiplication. Forφ1,φ2 ∈Γ(W),φ∈Γ(W)
D2φ=−div(∇φ) +S4φ,
< D2φ, φ2>=S4 < φ1, φ2>+<∇φ1,∇φ2>−δ <∇φ1,∇φ2>, where<∇φ1, φ2>is the 1-formv7−→<∇vφ1, φ2> .
(1.3) Let M be a closed, oriented Riemannian 4-manifold. By Hirzebruch and Hopf(1958), there is a spinc structure onM. Let a spinc structure onM be given.
Write W+ and W− for the associated spinc bundles. Then W+ is a u(2)-bundle and Clifford multiplication yield a linear isomorphism
ρ: Λ+⊗C−→Sl(W+).
Let L denote the determinant of W+, a line bundle onM andA be a unitary connection inL. LetDA: Γ(W+)−→Γ(W−) be the Dirac operator on the spinc bundle, and FA+ be the self-dual part of the curvature of A. Then we have the Weitzenb¨ock’s formula of the twisted Dirac operatorDA:
DA∗DAφ=∇A∗∇Aφ+S4φ−12ρ(FA+)φ.
2 Seiberg-Witten Invariant
Let M be a compact oriented smooth 4-manifold. Let W+ and W− be the associated spinc bundles for a given spinc structure onM. There is a pairing
σ:W+×W+−→Sl(W+)
modeled on the mapC2×C2−→Sl(2) given by (v, w)7−→i(vwt)0where 0 means the trace free part.
LetLbe the determinant ofW+, i.e., a line bundle onM. The Seiberg-Witten equations are the following pair of equations for a unitary connection A onLand a section φofW+ :
DAφ= 0 ρ(FA+) =iσ(φ, φ).
The gauge groupC∞(M, S1) acts on the Seiberg-Witten configuration space of all pairs (A, φ) via its action as scalar multiplication onφandA·g=g−2A(g2) = A+ 2g−1dgfor any g∈C∞(M, S1).
Theorem 2.1. The gauge group C∞(M, S1)acts on the solutions of the Seiberg- Witten equations. More precisely, if(A, φ)satisfies
DAφ= 0 ρ(FA+) =iσ(φ, φ),
then for any g∈C∞(M, S1),(A, φ)·g= (A·g, φ·g)satisfies DA·g(φ·g) = 0
ρ(FA·g+) =iσ(φ·g, φ·g).
We write M(L) for the moduli space, the quotient of the set of solutions by the gauge group. We call a solution irreducible if φ is not identically zero. Such solutions form free orbits of the gauge group. While the reducible so;utions,φ≡0, have stabilizerS1. If the homology groupH2(M, Z) has no 2-torsion, then the spinc structure is entirely determined by the topology ofL, which may be any line bundle whose first Chern class is an integral life of the second Stiefel-Wittney classω2. In
general, when L is fixed, the spinc structures with determinant L are a principal spare for the 2-torsion part ofH2(X, Z).
Theorem 2.2. Any solution (A, φ) of the Seiberg-Witten equations satisfies the C0 bound |φ|2 ≤ max {0,−s} at the points where |φ| is maximum. Here s is the scalar curvature ofM.
Proof. The Weitzenb¨ock formula for the Dirac operatorDA reads DA∗DAφ=∇A∗∇Aφ+s4φ−12ρ(FA+)φ.
At the maximum|φ|we compute
0≤ 4|φ|2= 2<∇A∗∇Aφ, φ >−2<∇Aφ,∇Aφ >
≤2<∇A∗∇Aφ, φ >
=−s
2 < φ, φ >+< iσ(φ, φ)φ, φ >
=−s
2|φ|2−1 2|φ|4.
In the last line we use the model ofσ. If |φ|2 is non-zero at the maximum, we may divide by|φ|2to obtain the bound|φ|2≤ −s. 2
The one aspect of Seiberg-Witten theory that differs from Donaldson theory is that there is no need for an Uhlenbeck-type compactification.
In Seiberg-Witten theory, there is an elliptic bootstrapping argument based on the fact that the solution (A, φ) of the Seiberg-Witten equations satisfies uniform boundedness. In Donaldson theory, there is conformally invariant for anti-self-dual equations in dimension 4.
Theorem 2.3. If (Ai, φi) is a sequence of solutions of SW-equations, then there is a subsequence (Ai0, φi0) and gauge transformations gi0 such that the sequence gi0 ·(Ai0, φi0)converges inC∞. Thus M(L)is compact.
3 Some Applications of Seiberg-Witten Invariant.
Consider Donaldson’s theorem asserting that the Donaldson invariants vanish for a connected sumX =X1]X2of 4-manifoldsX1andX2which each hasb2+>0.
The theorem is proved by considering a metric onX =X1]X2in whichX1andX2
are joined by a long neck of the form S3×I, where I is an interval in R1. Take a metric on the neck to be the product of the standard metric onS3 and a metric that assigns lengthttoI, and consider the Seiberg-Witten equations on this space.
For t−→ ∞, any solution of the Seiberg-Witten equations will vanish in the neck because of the positive scalar curvature ofS3. This follows from Theorem 2.2. This lets one define aU(1) action on the moduli spaceM(L) by gauge transforming the solutions on X2 by a constant gauge transformation, leaving fixed the data onX1. A fixed point of thisu(1)-action would be a solution for whichφvanishes onX1or onX2. But sinceX1andX2both haveb2+>0, there is no such solution if generic metric are used on the two sides.
Theorem 3.1.(Vanishing Theorem)
If a smooth closed 4-manifoldX is a connected sumX =X1]X2 of 4-manifoldsX1
andX2 which each hasb2+>0, then all Seiberg-Witten invariants ofX vanish.
Theorem 3.2.(Kronheimer-Mrowka)
Let Σ be an oriented 2-dimensional manifold smoothly embedded in CP2 so as to represent the same homology class as an algebraic curve degree d. Then the gauge g of Σsatisfiesg(Σ)≥ 12(d−1)·(d−2).
Theorem 3.2.(Traubes)
Let X be a closed symplectic 4-manifold with b2+ > 1 and a symplectic form ω.
Then the first Chern class of the associated almost complex structure on X has Seiberg-Witten invariant equal to±1.
Example.
The connected sumnCP2]CP2has no symplectic structure ifn >1.
References.
[1] Y. S. Cho, Seiberg-Witten invariants and 3- and 4-manifolds, Preprint.
[2] Y. S. Cho,Equivariant metric for smooth moduli spaces, Topology and its Ap- plications 62(1995), 77-85.
[3] Y. S. Cho,Cyclic group actions on Gauge theory, Differ. Geom. and its Appli- cations , (to appear).
[4] Y. S. Cho,Finite group actions on the moduli space of self-dual connections (I), Trans. A.M.S.323(1991), 233-261.
[5] S. K. Donaldson,The orientation of Yah-Mills moduli space 4-manifold topology, J. Differential Geometry 26(987), 397-428.
[6] M. Gromov,Pseudo-holomorphic curves in symplectic manifolds, Invent. Math.
(82) (1985), 307-347.
[7] P. Kronheimer and T. Mrowka,The genus of embedded surfaces in the projective plane, Math. Research Letters1(1994), 794-808.
[8] D. Kotschick, S. Morgan and C. Taubes, Four-manifolds without symplectic structures but with nontrivial Seiberg-Witten invariants, Math. Research Letters2 (1995), 119-124.
[9] C. H. Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Re- search Letters1(1994), 809-822.
[10] E. Witten, Monopoles and four-manifolds, Math. Research Letters 1(1994), 769-796.
